A Dilworth Decomposition Theorem for λ Suslin Quasi-Orderings of R

Publisher Summary This chapter focuses on Dilworth decomposition theorem for λ-suslin quasi-orderings of ℝ. The chapter shows that if ≤ is a Borel quasi-ordering of ℝ with no perfect set of incomparable elements, then (1) ℝ = ∪ n∈ω X n, where each X n is ≤ −linearly ordered and Borel and (2) there is a strictly ordered preserving map F : (ℝ,≤)→ (2 α , lex ) for some α 1 . The chapter proves some lemmas for separation. These are analogous to Suslin's theorem.